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Kernel and cokernel are mutually inverse order-preserving correspondences between subobjects and quotient objects

Statement

Fix an object $A$ in an abelian category. Sending a subobject representative $m : M \to A$ to its cokernel class $[coker (m)]$ , and sending a quotient representative $q : A \to Q$ to its kernel class $[\ker (q)]$ , defines mutually inverse order-preserving bijections between the subobjects of $A$ and the quotient objects of $A$ .

Facts & Assumptions

Given: An object $A$ in an abelian category.

[L1]

Subobjects and quotient objects are mutual-factorization classes with the opposite order conventions (Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms, Subobjects and quotient objects form oppositely oriented partially ordered collections).

[L2]

Mutual factorization is the correct representative-independent equality relation on subobjects and quotient objects (Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it).

[L3]

Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel (Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel).

Proof

technique · direct

1.1

Let $m : M \to A$ be monic, let $c_{m} : A \to Q_{m}$ be its cokernel, and let $k_{m} : K_{m} \to A$ be the kernel of $c_{m}$ . Since $c_{m} m = 0$ , the monomorphism $m$ factors through $k_{m}$ . Conversely, [L3] says that $m$ is itself a kernel of $c_{m}$ , so $k_{m}$ factors through $m$ . Thus $[k_{m}] = [m]$ . The dual argument shows that for every epic $q$ one has $[coker (\ker q)] = [q]$ .

L2L3

1.2

If $[m] \leq [n]$ , then $m = n u$ for some $u$ . Since $coker (n) n = 0$ , one has $coker (n) m = 0$ , so the cokernel universal property of $m$ makes $coker (n)$ factor through $coker (m)$ . By the quotient-order convention in [L1], this is exactly $[coker (m)] \leq [coker (n)]$ . The kernel assignment preserves the order dually.

L1L3

2.1

Step 1.1 proves that the two assignments are mutually inverse on classes, and step 1.2 proves that both preserve the stated orders. So kernel and cokernel are mutually inverse order isomorphisms between subobjects and quotient objects.

step 1.1step 1.2∎

Depends on

Used by

The quotient by a subobject is independent of the chosen representing monomorphism Theorem

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Junhan Tan, The Freyd-Mitchell Embedding Theorem, Theorem 2.3 (standard reference, not scraped)